Algorithm and implementation system for measuring impedance in the d-q domain

ABSTRACT

A controller and infrastructure for an impedance analyzer measures responses to perturbations to respective phases of a multi-phase system at an interface between stages thereof (which may be considered as a source and load in regard to each other), such as a multi-phase electrical power system, to determine a transfer function for each phase of the multi-phase system from which the impedance of each of the source and load can be calculated, particularly for assessing the stability of the multi-phase system.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims benefit of priority of U.S. ProvisionalApplications 61/532,588 filed Sep. 9, 2011 and 61/535,398, filed Sep.16, 2011, both of which are hereby incorporated by reference in theirentireties. This application is also related to U.S. patent applicationSer. No. 13/608,213, filed Sep. 10, 2012, which is also incorporated byreference in its entirety.

FIELD OF THE INVENTION

The present invention generally relates to a methodology and apparatusfor direct measurement of impedance of an AC power source and powerconverter and, more particularly, to measurement of impedance ofportions of a power circuit at interfaces therebetween and which may betime-varying and/or non-linear; which measurements may be conductedon-line while the power circuit is delivering power to a load.

BACKGROUND OF THE INVENTION

Traditionally, electrical power has been produced by large,geographically separated facilities and transmitted as high-voltagealternating current to other locations. These large Power generationfacilities are connected through a network, sometimes referred to as apower grid such that power produced by locations having excess powergeneration capacity can be diverted to areas where loads may beparticularly large at any given time. In the proximity of loads, thevoltage is generally reduced in stages and further distributed until thelocation of various loads is reached. The high voltage used fortransmission over long distances allows currents and resistive losses tobe reduced while using cables of reduced conductor material content. Useof alternating current allows the reduction of voltage by the use oftransformers. Alternating current can also be directly used by manycommon and familiar loads such as household appliances, pumps usingelectric motors and the like.

However, many familiar loads are principally based on electroniccircuits which are rapidly increasing in number and power requirementsand the proportion of the load of many other devices that is presentedby electronics (e.g. processor controlled appliances) is also increasingrapidly. Many new devices such as electrically powered vehicles are alsobeing introduced. Most of these types of loads require direct current(DC) power. Additionally, environmental concerns have encouraged thedevelopment of local power generation and/or storage systems in manylocations to serve local “islands” or groups of customers where powerdistribution can be provided as either AC or DC current. Power storagemust generally be provided with DC power. Therefore, the need forconversion between AX and DC power is proliferating rapidly at thepresent time and is likely to accelerate.

Power converters are, by their nature, non-linear and their dynamicbehaviors are coupled with those of the load from which they receivepower or the source providing power through them. As a consequence, manypower electronics systems will require control in order to provide aregulated output. However, provision of such regulation causesadditional phenomena that have not been previously observed orconsidered to be of importance, including but not limited to issues ofstability.

Specifically, a power converter under regulated output control exhibitsnegative incremental impedance characteristics at its input. That is, inthe case of converters regulating voltage (to a different form of thatof the source), the current consumed by them is inversely proportionalto voltage variations of the source in order to maintain a constantpower flow to the load. This is the inverse behavior of resistive loads,whose current consumption is directly proportional to voltage variationsof the source. Consequently, the small-signal response at a givenoperating point, corresponding to the linearization of the converter atsuch point, presents negative phase. As is well recognized in the art,negative impedance can result in instabilities and possibly oscillatorybehavior of the circuit with detrimental effects to the system wherethey operate.

While the extensive power grid can tolerate many of these behaviorssince the effect of converter behavior is small compared to the size ofthe system, such behaviors cannot be tolerated by smaller systems whichhave their own, relatively small capacity power source and are notconnected to the effectively infinite power grid. Examples of suchsmaller systems are aircraft, water-borne vessels, hybrid electricvehicles and small power plants (e.g. wind turbines or solar collectorfarms) serving individuals or small “islands” of customers. Otherexamples of circumstances where unstable behavior may occur areinstances where electrical loads are connected through equipment such asfrequency changers (AC/AC converters), AC/DC converters and other typesof hardware. Vehicular systems also operate at higher line frequenciesthan the line frequencies traditionally used for power distribution andpresent other phenomena and challenges in regard to control.

Systems which can potentially exhibit unstable behaviors are becomingprevalent due to proliferation of systems such as are discussed aboveand, further, by shifting functions previously performed mechanically orhydraulically to electrically powered functions. Accordingly, it isimperative that potential instabilities be made predictable and avoidedin the design of such systems. Therefore, stability of electricalsystems has been a subject of substantial interest and study in recentyears; yielding some solutions for DC systems such as DC/DC converters.However, there are issues not seen in DC systems which are present in ACsystems and multi-phase AC/DC systems, in particular, which are referredto as multi-variable systems. While some progress has been made inregard to determining stability or forbidden operating conditions ofmulti-phase AC/DC systems, the analysis has been extremely complex andburdensome and has, in general, led to excessively conservative designsand operating parameters.

Many authors have also suggested algorithms to extract parameters fromthe system to fit a predetermined system model. While this approach isnot black-box impedance measurement, as it is based on a known model ormathematical description of the system, it is noteworthy as it isanother technique used to acquire actual data to fit a model of thesystem. It is however, not as accurate as individual point measurements.

Additional work has been done on the capture of dynamics via artificialrecurrent neural networks in the d-q coordinate system. These methodsinject noise into the system and learn and record the response of thesystem dynamics to the noise. The system input-output relationship canthen be learned from the response. This approach has an advantage inthat if the network is trained properly it can filter out exogenousnoise from the measurements. Although this approach does capture thedynamics of the system, it does not address extraction of the impedancefrom the dynamic results, and focuses mostly on the network itself. Itshould be noted that all three of these techniques are only simulated.There is no discussion of the error of the approximation.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide a anapparatus for measuring impedance of respective phases of a multi-phaseAC supply, power converter stage(s) and/or load at an interfacetherebetween for purposes of evaluating stability of the combinationthereof.

In order to accomplish these and other objects of the invention, anapparatus for measuring impedances of respective phases of a multi-phaseelectrical power system at any system interface between stages of themulti-phase electrical power system is provided comprising a phaselocked loop for aligning a frame of reference with an input power vectorand a plurality of angles, a sweep generator for applying perturbationsto respective phase of the interface over a range of frequency, insequence, to the multi-phase electrical power system, voltage andcurrent sensors for measuring amplitude and phase of the voltage andcurrent responses to respective perturbations, and a computer to computeimpedances of respective phases of the interface from phase andamplitude of voltage and current responses to the perturbations over therange of frequency of respective perturbations.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, aspects and advantages will be betterunderstood from the following detailed description of a preferredembodiment of the invention with reference to the drawings, in which:

FIG. 1 is a schematic illustration of on-line measurement of impedanceat an exemplary interface of a three-phase power converter system,

FIG. 2 is a schematic illustration of an equivalent circuit of FIG. 1represented in the d-q domain,

FIG. 3 is a schematic depiction of a DC power conversion system underperturbation,

FIG. 4 is a functional block diagram of an analyzer in accordance withthe invention,

FIG. 5 illustrates a suitable exemplary architecture for performing theanalyzer algorithm in accordance with the invention,

FIG. 6 illustrates a preferred form of a prototype analyzer inaccordance with the invention,

FIG. 7 is a schematic diagram of an exemplary and generalized systemunder test with the analyzer inserted,

FIG. 7A is a diagram useful for understanding measurement sweeps inaccordance with the invention,

FIG. 8 graphically illustrates experimentally measured direct-channelimpedances of an exemplary test circuit,

FIG. 9 graphically illustrates experimentally measured cross-channelimpedances of the same exemplary test circuit, and

FIG. 10 summarizes the development of impedance and admittancemeasurements from the measurements made by an embodiment of theinvention.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT OF THE INVENTION

Referring now to the drawings, and more particularly to FIG. 1, there isshown a schematic diagram of a generalized there-phase power systemhaving an interface between a source and a load. It should be understoodthat any juncture between portions of a power converter system willprovide an interface between “upstream” circuits and “downstream”circuits and that the former functions as a source for the latter andthe latter may be considered as a load for the former. In the schematicdiagram of FIG. 1, current sources i_(Pa), i_(Pb) and i_(Pc) are capableof injecting current into and thus perturbing the respective phases ofthe power converter circuit. The source and load impedances (e.g.Z_(sa)(s) and Z_(la)(s) for phase a) of respective phases can thus bederived from the resulting voltages and upstream and downstream currents(e.g. voltage v_(an), v_(sa) and v_(La) and currents i_(sa) and i_(La)for phase A).

The above-incorporated U.S. patent application Ser. No. 13/608,213discloses a simplified methodology for studying power circuit stabilityin which the multi-variable problem is reduced in several ways to beamenable to relatively much more simple analysis. Specifically, in thatapplication, it was demonstrated that stability of a power circuit isdetermined by the scalar return-ratio across the d-q frame active powertransfer channel which is the d-d channel for a V_(q)=0 alignment of thed-q frame. The multi variable stability problem of multi-phase AC powersources or loads is reduced to the scalar case and can be analyzed bythe single input, single output (SISO) standard Nyquist stabilitytheorem, as distinct from requiring use of the generalized Nyquiststability theorem or the multi-variable Nyquist stability theorem whichare far more complex and burdensome and lead to excessively conservativeevaluations as applied to power converters. Further, it was demonstratedthat there is no need to know all the system dynamics but only theimpedance characteristics at each interface of the power converter. Suchimpedance characteristics can be determined through circuit analysisbut, preferably, can be more conveniently derived through directmeasurement. However, since it is a requirement for valid stabilityanalysis that such a measurement be conducted on-line while rated poweris being delivered to a load, significant requirements are imposed onthe required instrumentation which the invention fully answers.

The power system thus interfaces devices used to control power flow inorder to provide power to the load, such as three-phase powerconverters. Analysis of these devices and systems can be performed atmultiple levels, ranging from power flow and power quality all the wayto models of the solid state semiconductor devices in the convertersthemselves. Appropriate models are chosen based on the level of analysisto be performed. Since this invention focuses on measuring impedance asa function of frequency, the models will be small signal modelsrepresenting the converter and subsystem(s) behavior at a givenoperating point.

Although these systems have been simplified by such models, they remainchallenging to analyze, often providing multiple stability points (andtherefore regions of attraction, and a series of other nonlinearphenomena, ranging from bifurcation to limit cycles and chaos.Furthermore, such nonlinear systems operate on a nominal trajectory insteady-state operation, nominally given by

$\begin{matrix}{{{v_{a}(t)} = {V_{m}{\cos \left( {\omega \; t} \right)}}}{{v_{b}(t)} = {V_{m}{\cos \left( {{\omega \; t} - \frac{2\; \pi}{3}} \right)}}}{{v_{c}(t)} = {V_{m}{\cos \left( {{\omega \; t} + \frac{2\; \pi}{3}} \right)}}}} & (1)\end{matrix}$

making them non-stationary systems with periodic tendencies.

To simplify analysis, attempts have been made and have given rise toconvention, to map this non-stationary system and its components to onewhich is stationary, mitigating (but unfortunately, in practice, notcompletely eliminating) the non-autonomous nature of such systems. Arotating coordinate system can be defined that matches the frequency ofrotation of the voltage vector, making the voltage appear stationary inthat frame of reference (referred to hereinafter simply as “frame”).This transformation is shown by

$\begin{matrix}{\begin{bmatrix}{v_{d}(t)} \\{v_{q}(t)} \\{v_{0}(t)}\end{bmatrix} = {{\sqrt{\frac{2}{3}}\begin{bmatrix}{\cos \left( {\omega \; t} \right)} & {\cos \left( {{\omega \; t} - \frac{2\; \pi}{3}} \right)} & {\cos \left( {{\omega \; t} + \frac{2\; \pi}{3}} \right)} \\{- {\sin \left( {\omega \; t} \right)}} & {- {\sin \left( {{\omega \; t} - \frac{2\; \pi}{3}} \right)}} & {- {\sin \left( {{\omega \; t} + \frac{2\; \pi}{3}} \right)}} \\\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{bmatrix}}\begin{bmatrix}{v_{a}(t)} \\{v_{b}(t)} \\{v_{c}(t)}\end{bmatrix}}} & (2)\end{matrix}$

For the systems of interest, the third component, known as the 0-axis,can be ignored. In effect, if the system is balanced, this axis iseffectively zero. For linear analysis of nonlinear systems, it isnecessary to have an operating point upon which to performlinearization. When the system is unbalanced, this operating pointdisappears when using the map described above, and the system cannot belinearized. Thus classical stability analysis becomes difficult withoutfurther tools. If the three voltages follow the trajectory specified inequation (1) then the resulting vector in the d-q frame, calculated byapplying (2) will be

$\begin{matrix}{\begin{bmatrix}{v_{d}(t)} \\{v_{q}(t)} \\{v_{0}(t)}\end{bmatrix} = {{\sqrt{\frac{2}{3}}\begin{bmatrix}\frac{3}{2} \\0 \\0\end{bmatrix}}V_{m}}} & (3)\end{matrix}$

Systems of loads and sources, although stable individually, may becomeunstable when they are interconnected. Stability in the d-q frame hasbeen explored previously for systems whose impedance is known. Based onthe analysis of these methods, it is possible to predict whether or notthe interconnection of two power electronics subsystems operating at anoperating point will produce a stable system. In order to apply thesemethods in practice, it becomes necessary to be able to measure theimpedance to which to apply the criteria respectively proposed therein.

Given a three-phase A-B-C system (simplified in FIG. 1 to awye-configuration for ease of understanding), a shunt current source isplaced at the point of measurement as shown in FIG. 1. This system, whentransformed to the D-q domain may be represented as shown in FIG. 2. Asingle port balanced time-invariant network represented in the D-qdomain may be described by the following transfer function matrix:

$\begin{matrix}{\begin{bmatrix}{v_{d\; 1}(s)} \\{v_{q\; 1}(s)}\end{bmatrix} = {\begin{bmatrix}{Z_{Ldd}(s)} & {Z_{Ldq}(s)} \\{Z_{Lqd}(s)} & {Z_{Lqq}(s)}\end{bmatrix}\begin{bmatrix}{i_{{Ld}\; 1}(s)} \\{i_{{Lq}\; 1}(s)}\end{bmatrix}}} & (4)\end{matrix}$

Due to the stationary nature of the system, it can be assumed that thetransfer function between different inputs and outputs can be measuredsequentially or simultaneously without any change in the results, andthat measurements can be repeated as many times as necessary, retrievingthe same transfer function each time. If only one side (e.g. load orsource) of the system was being measured, current could simply beinjected into the D-axis and the corresponding voltage componentsmeasured while keeping the Q-axis zero. As the injection must be in ashunt configuration to achieve an on-line measurement, however, this isnot the case.

The perturbation is a vector and, due to the shunt configuration, thatvector will split when the perturbation reaches the point of commoncoupling. It is assumed, however, that when a current injection is madeto perturb at a different angle, that the load and source current andvoltage vector perturbation will rotate by that same angular difference.

There are several challenges to overcome to accomplish such ameasurement. Nearly all systems are nonlinear, and as such, require anoperating point in order to take a measurement. This requires the systemto be operating during the measurement. As most of these converters aredesigned for a high power level, it precludes the use of mostcommercially available equipment used to measure impedance viatraditional means. Such equipment can take highly accurate measurementsbut only at very low power levels. For linear, time-invariant, balancednetworks, an operating point is not required, but for nonlinear systems,unless the impedance can be transformed a posteriori, it is necessaryfor the system to be at the system operating point. This is arequirement for nearly all systems that involve power electronicsequipment.

Such a restriction gives rise to several challenges in implementing ameasurement system for such a measurement subsystem. The first is theability to induce a perturbation into a system. Such an injection mustbe supplied at a reasonable magnitude (e.g. 1-2% of system rated power)in order to perturb the system, and the injection equipment must be ableto operate with other power sources active in the systems which aresignificantly larger than the injection. Furthermore, unlike traditionalanalyzers, the analyzer in accordance with the invention must measurethe impedance in an artificial frame of reference that does notphysically exist. There are no d- and q-axis terminals to which one mayconnect a sensor, and such a reference frame must be derived via realtime processing.

The nonlinear nature of these systems requires them to be running whenmeasured, and therefore most attempts to inject a perturbation includingthose reported in DC/DC converter literature, have provided methods forsuch. Without the challenge of a rotating coordinate system, logic orimpedance analyzers can be directly connected to amplifiers or couplingnetworks which couple the perturbation to the system. This is shown inFIG. 1 where the perturbation is injected by the shunt current sourcesas illustrated. This Figure should be understood to illustrate a genericthree-phase system where the source supplies a load, where both thesource and load are represented by an equivalent Thevinin circuitrepresentation. The input impedance and impedance of the converter maybe measured in this case by measuring the current and voltage responseto the perturbation current.

Referring now to FIG. 3, to create a perturbation at an interface, aseries voltage source or current source may be used or a series or shuntconfigured impedance may be used. These devices modify the systemcurrents and voltages in order to create a perturbation. It should benoted that the location where the system is perturbed is not necessarilythe same location where the resulting perturbation is measured. That is,i_(p2) may be used to create a perturbation while i₁ and v₁ are beingmeasured. The sources shown in FIG. 3 are not in the system, but areplaced there for the purpose of measuring impedances within the systemvia injection hardware.

When measuring impedance or admittance, a small signal phenomenon, aninput (current or voltage) and an output (voltage or current) must bemeasured. For a linear system (or a nonlinear system that is linearizedabout an operating point), the input and output signal components, bothat the same frequency, are related by a transfer function which definesthe gain and phase shift at that frequency.

It has been reported that a power converter was used to generate aperturbation into the system on all three phases. A voltage sourceinverter was attached to the system, and was provided power on the DCside from an external source. The converter was shunt connected to thepower system. Activating the converter switches allowed the converter toinject current into the system. It has been theoretically shown thatthis technique may be performed using an active filter but such atechnique has not been demonstrated. The results were simulated.

A wound-rotor induction machine can be used to inject a perturbationinto the system. DC current is injected into the machine and the machineis allowed to synchronize with the system, after which the perturbationcan be injected on top of the DC signal. The machine injects onto allthree phases as it rotates.

A third technique modulates a three-phase shunt-connected resistiveimpedance (done with a three-phase chopper circuit). This injection ismade smoother with the addition of a series inductor. A powersemiconductor switch shorted one resistor to create the modulation. Asimilar method to inject a perturbation was also created that modulatedan impedance only between two of the three phases.

Another injection method that has been proposed is series voltageinjection. However, this is less practical due to the large currentspresent in the system.

The converter-based perturbation methods above have been directlyconnected to the system. If isolation is desired or the electronics usedare insufficient to inject a signal of proper magnitude due to thelimitations of the electronics involved, the use of a transformer may bewarranted. Using a transformer, however, imposes additional restrictionson the injected frequency content, as will be discussed.

While injection itself is a challenge, a second challenge involves thepresence of exogenous signals in the network during measurement. Sincethe network is nonlinear, it is measured during its operation. As such,there are other exogenous signals present due to the system's operation.These include, but are not limited to, line frequency harmonics,switching ripple, low frequency modulation effects, zero crossingdistortions due to non-ideal behavior of diodes and diode rectifierbridges in the system, load-source interactions, and others. Whileattempts have been made to mitigate these effects, they still prevail inmany systems.

In the d-q frame, an alignment is chosen which defines the frame. Ingeneral, either the d-axis or the q-Axis is aligned to the rotatingvoltage vector. If the d-q frame is aligned to a different angle, themeasured impedance may also change. A property of showing no change whenchanging the alignment angle of the d-q frame to the rotating voltagevector is referred to as isotropism or rotational invariance. If theimpedance of a system is dependent on the alignment of the d-q frame theimpedance is called anisotropic. It is therefore necessary foranisotropic systems that the d-q coordinate system is aligned properlywith the variable of interest. In accordance with the invention, it ispreferred, but not required, that the d-q frame will be aligned with thed-axis such that the q-axis voltage component is centered on zero forsimplicity. Such an alignment will be assumed in the followingdiscussion.

This alignment is usually achieved via a phase locked loop (PLL) thatcontrols the reference frame angular velocity until it aligns with therotating voltage vector. However, if the voltage vector has harmonics ornoise, or if there are imbalances in the system created by the systemitself or by the injected perturbation, the PLL will have a reaction toit, and the frame will no longer be rotating at exactly a constantfrequency. Instead of a PLL, a low pass filter on the line voltage hasbeen attempted but will be even more significantly affected by theseharmonics and other exogenous signals as the basis voltage will containlow frequency perturbation signals. To date, no discussion of the PLL ispresented in researched literature for the purpose of impedancemeasurement. Nonetheless, several PLL designs have been found to berobust to the presence of system imbalance.

As there are two channels, d and q, the impedances are expressed asmatrices, and there exists coupling between the load and sourcesubsystems represented by these matrices. A perturbation on thed-channel can cross-couple to the q-channel output, which can theninteract with the load, and again cross-couple to produce a voltageresponse back on the d-channel. This interaction makes the impedanceappear to include the load, and is a result of having a multi-variablesystem. The solution must decouple this interaction.

An additional challenge that arises when building a measurement systemis the ability to verify that the measurements are correct. In the caseof linear networks it is possible to derive the expected d-q impedancesgiven symmetric, linear, time-invariant impedances of each phase instationary coordinates. It should be noted that no published result seenthat did conduct experimental work verified the full impedance they weremeasuring against known impedances by measuring them using dedicatedequipment. The closest to this was a low order parametric modelconstructed using nominal parameters of a load inductance andresistance.

For other systems, such as voltage source inverters, an approximatemodel is well-known representing the system dynamics under idealconditions. However, this approximate model is derived from idealswitching behavior, and is not without assumptions. For verificationpurposes, it is important to know that the model is accurate andrepresents the true behavior of the converter despite the presence ofother time-varying and nonlinear behavior such as converter dead-timeand the potential discontinuous conduction of each phase around the zerocrossing.

The need to know the system impedance has been made apparent based onsystem stability requirements, which have recently become importantbased on the ever-increasing demands of equipment with destabilizingeffects on their host systems. These motivations are increased by theincreasing number of small systems with limited power generationcapability and the increased transition of former hydraulic andmechanical systems to electrical power, as alluded to above.

Previous attempts to measure three-phase impedances have beenincomplete. Results presented by these previous attempts do not provideconfidence that the measurements are being performed in an approachacceptable for all load types, especially ones containing powerconverters. Reasons for this lack of confidence include a series ofissues related to d-q frame alignment, nonlinear load behavior,multi-channel power flow, and a range of exogenous signals preventingsuccessful and complete measurement.

Furthermore, no published work attempts to characterize theirmeasurement system for accuracy. As the objective is to formulate ameasurement instrument, it is essential to know its limitations,operational boundaries and capabilities to avoid accepting incorrectmeasurements. Additional work is required in order to understand thesecharacteristics and capabilities.

In order to address these problems, certain features of these problemsmust be individually addressed. First is the D-q frame. If measurementsare to be taken in the D-q frame, it is necessary that a stablereference frame is established prior to measurement at the desired anglewith respect to the system voltage or current and that does not varyduring any measurements taken.

Any phase locked loop may work for the application given that it canhave a variable bandwidth which can be changed during runtime.

However, due to small imbalances, a three-phase PLL is selected thatdecouples the positive and negative sequences of the voltage with whichit is synchronizing. Keeping in mind the algorithm will be implementedin software, a version of the referenced PLL, once tailored to thespecific voltage sequences, can be simplified. Once the reference frameis defined via the PLL, it is possible to analyze the system dynamics inthis frame by introducing perturbations into the system.

Given a current perturbation, i_(p)(s), into the system, theperturbation will enter and split, some going to the load and some goingto the source as illustrated in FIG. 1. For compactness of notation, thefollowing are defined:

$\begin{matrix}{{v(s)} = {{v_{S}(s)} = {{v_{L}(s)} = \begin{bmatrix}{v_{d}(s)} \\{v_{q}(s)}\end{bmatrix}}}} & (5) \\{{{i_{S}(s)} = \begin{bmatrix}{i_{S\; d}(s)} \\{i_{S\; q}(s)}\end{bmatrix}}{{i_{P}(s)} = \begin{bmatrix}{i_{P\; d}(s)} \\{i_{P\; q}(s)}\end{bmatrix}}{{i_{L}(s)} = \begin{bmatrix}{i_{L\; d}(s)} \\{i_{L\; q}(s)}\end{bmatrix}}} & (6)\end{matrix}$

Accordingly, the impedances can be defined as:

$\begin{matrix}{{{Z_{L}(s)} = \begin{bmatrix}{Z_{Ldd}(s)} & {Z_{Ldq}(s)} \\{Z_{Lqd}(s)} & {Z_{Lqq}(s)}\end{bmatrix}}{{Z_{S}(s)} = \begin{bmatrix}{Z_{Sdd}(s)} & {Z_{Sdq}(s)} \\{Z_{Sqd}(s)} & {Z_{Sqq}(s)}\end{bmatrix}}} & (7)\end{matrix}$

From FIG. 2, it follows that

i _(s)(s)+i _(p)(s)=i _(L)(s)−Z _(s)(s)⁻¹ v(s)+i _(p)(s)=Z _(L)(s)⁻¹v(s)  (8)

Giving:

$\begin{matrix}{{\left\lbrack {{Z_{L}(s)}^{- 1} + {Z_{S}(s)}^{- 1}} \right\rbrack^{- 1}{i_{P}(s)}} = {v(s)}} & (9)\end{matrix}$

Thus, if estimates of the values of Z_(s) and Z_(L) are available, anestimate can be made as to how much perturbation must be injected inorder to achieve a desired voltage perturbation to be measurable beyondthe quantization non-linearities introduced by the load, source, andmeasurement system.

Similarly, it can be shown that the current will split between the loadand source as shown below:

v(s)=v _(s)(s)=v _(L)(s)  (10)

[Z _(L)(S)⁻¹ +Z _(S)(S)⁻¹]⁻¹ i _(P)(S)=−Z _(S)(S)i _(S)(S)=Z _(L)(S)i_(L)(S)  (11)

From equations (10) and (11), the current to each side can be directlycalculated as:

Z _(S)(S)⁻¹ [Z _(L)(S)⁻¹ +Z _(S)(S)⁻¹]⁻¹ i _(P)(S)=i _(S)(S)

Z _(L)(S)⁻¹ [Z _(L)(S)⁻¹ +Z _(S)(S)⁻¹]⁻¹ i _(P)(S)=i _(L)(S)  (12)²)

In this regard, it is also helpful to recognize that

i _(S)(S)=−Z _(S)(S)⁻¹ Z _(L)(S)i _(L)(S)  (13)

Thus, whenever one impedance is much greater than the other impedance,it will be difficult to measure the current response related to thatside, as most of the current will flow in the opposite direction,leading to measurement range restrictions as applied to the currentsensors and the associated A/D converters, which must measure the largesignal response while being able to accurately measure the small signalresponse. Based on the estimated value of the impedances, this can beused to calculate the necessary perturbation current or voltage value inorder to cause a disturbance in the system of sufficient magnitude.

Many commercially available instruments are available to measure thetransfer function between an input and an output, and as such, it ispreferred to utilize such equipment in the practice of the invention.Gain and phase analyzers have the ability to reject components of thesignal which are not closely related to its own perturbation. Regardlessof the technique or instrument used, the purpose of this block (alsoshown in FIG. 4) is to measure the gain from a single input to a singleoutput.

As only one input-output relationship can be measured at a time and thecomplete system is represented by a multi-input, multi-output transferfunction (the impedance), it will be necessary to make multiple sweepsto fully characterize the impedances of multi-phase systems, where asweep refers to the continuous excitation of the system withperturbations of, for example, increasing frequency value. Furthermore,as there is a coupling between the load and source subsystems, oneinjection cannot easily be set to be identically zero. As can beobserved in equation (10), both the load and source impedances arepresent in both equations, and they must be decoupled if one is toisolate a load or a source subsystem from others.

As schematically shown in FIG. 4, this is accomplished by injecting oni_(Pd) and i_(Pq), and obtaining responses on i_(Sd)(s), i_(Sq)(s),i_(Ld)(s), i_(Ld)(s), v_(d)(s), and v_(q)(s) as described earlier.Similarly, rotating the perturbation vector, i_(P) so that i_(Pd) andi_(Pq) have a new and different proportion, there will be a second setof linearly independent responses on the vectors i_(s), i_(L) and v. Oneach of the measurements, these signals are measured as three-phasevariables and converted into the d-q frame in the signal processingelement 41 as shown in FIG. 4. The corresponding d-q terms are then sentout to the gain/phase analyzer 42 as signal A, which ultimately measuresand calculates the response signals of interest, that is, transferfunctions. The gain/phase analyzer 42 similarly generates theperturbation signal_(Rfout) that is converted from the d-q frame into athree-phase variable by the signal processing block 41 prior to itsinjection into the system under test 43 through amplifiers.

In the following equations, let the subscripts 1 and 2 denote twoindependent perturbation angles. When measuring the load, injection intoi_(Pd) gives

$\begin{matrix}{\begin{bmatrix}{v_{d\; 1}(s)} \\{v_{q\; 1}(s)}\end{bmatrix} = {\begin{bmatrix}{Z_{Ldd}(s)} & {Z_{Ldq}(s)} \\{Z_{Lqd}(s)} & {Z_{Lqq}(s)}\end{bmatrix}\begin{bmatrix}{i_{{Ld}\; 1}(s)} \\{i_{{Lq}\; 1}(s)}\end{bmatrix}}} & (14)\end{matrix}$

Similarly, injecting a second time with a linearly independent injectionvector gives a rotated response:

$\begin{matrix}{\begin{bmatrix}{v_{d\; 2}(s)} \\{v_{q\; 2}(s)}\end{bmatrix} = {\begin{bmatrix}{Z_{Ldd}(s)} & {Z_{Ldq}(s)} \\{Z_{Lqd}(s)} & {Z_{Lqq}(s)}\end{bmatrix}\begin{bmatrix}{i_{{Ld}\; 2}(s)} \\{i_{{Lq}\; 2}(s)}\end{bmatrix}}} & (15)\end{matrix}$

Notice that the impedance has not changed as the system is the same. Theresponse has changed as dictated by the inputs and outputs. Theequations containing v_(d1) and v_(d2) can be regrouped as:

$\begin{matrix}{\begin{bmatrix}{v_{d\; 1}(s)} \\{v_{d\; 2}(s)}\end{bmatrix} = {\begin{bmatrix}{i_{{Ld}\; 1}(s)} & {i_{{Lq}\; 1}(s)} \\{i_{{Ld}\; 2}(s)} & {i_{{Lq}\; 2}(s)}\end{bmatrix}\begin{bmatrix}{Z_{Ldd}(s)} \\{Z_{Ldq}(s)}\end{bmatrix}}} & (16)\end{matrix}$

From equation (16), the impedances Z_(ldd)(s) and Z_(ldq) (s) can berespectively computed as:

$\begin{matrix}{{\begin{bmatrix}{i_{{Ld}\; 1}(s)} & {i_{{Lq}\; 1}(s)} \\{i_{{Ld}\; 2}(s)} & {i_{{Lq}\; 2}(s)}\end{bmatrix}^{- 1}\begin{bmatrix}{v_{d\; 1}(s)} \\{v_{d\; 2}(s)}\end{bmatrix}} = \begin{bmatrix}{Z_{Ldd}(s)} \\{Z_{Ldq}(s)}\end{bmatrix}} & (17)\end{matrix}$

Similarly, this technique can be reapplied to the q-axis voltages:

$\begin{matrix}{{\begin{bmatrix}{i_{{Ld}\; 1}(s)} & {i_{{Lq}\; 1}(s)} \\{i_{{Ld}\; 2}(s)} & {i_{{Lq}\; 2}(s)}\end{bmatrix}^{- 1}\begin{bmatrix}{v_{q\; 1}(s)} \\{v_{q\; 2}(s)}\end{bmatrix}} = \begin{bmatrix}{Z_{Lqd}(s)} \\{Z_{Lqq}(s)}\end{bmatrix}} & (18)\end{matrix}$

Transposing and stacking these equations yields the final form:

$\begin{matrix}{{\begin{bmatrix}{i_{{Ld}\; 1}(s)} & {i_{{Lq}\; 1}(s)} \\{i_{{Ld}\; 2}(s)} & {i_{{Lq}\; 2}(s)}\end{bmatrix}^{- 1}\begin{bmatrix}{v_{d\; 1}(s)} & {v_{d\; 2}(s)} \\{v_{q\; 1}(s)} & {v_{q\; 2}(s)}\end{bmatrix}} = \begin{bmatrix}{Z_{Ldd}(s)} & {Z_{Ldq}(s)} \\{Z_{Lqd}(s)} & {Z_{Lqq}(s)}\end{bmatrix}} & (19)\end{matrix}$

When measuring these responses, if the same filter (if used) is appliedto every measurement, then that filter will cancel as it can be factoredout of the matrix. Similarly, processing delay introduced in eachmeasurement will be eliminated. Thus, symmetrical linear elements suchas anti-aliasing filters present on both the current and voltagemeasurements will not influence the impedance calculation. It istherefore desirable to make all measurement channels to be as similar aspossible to eliminate distortion and maintain valid transfer functionmeasurements.

In the solution, it should be noted that the procedure does not directlymeasure ratios of voltage to current perturbation responses, but insteadmeasures voltages and currents with respect to a given repeatablereference, which remains the same for all the sweeps. Eight variablesare therefore measured (four currents and four voltages) per injection,and those injections occur at two different angles.

As the procedure measures only one transfer function at a time, it isnecessary to assume the measurements are repeatable, and that the systemhas extremely little time variation in the d-q domain. If the systemwere to be able to move its operating point during the measurement, itwould be impossible to sequentially measure the transfer functions, asthe state of the system would be different for each measurement.Similarly, the system must be operating at a constant frequency, andthat frequency should not change with the introduction of aperturbation.

More than two sets of vectors may be used as well to calculate themeasured impedances if multiple frequency sweeps are conducted measuringmultiple responses. This requires that the multiple sweeps are performedat the exact set of frequency points so that multiple measurements existat each of these points. This results in the construction of largermatrices as opposed to the two vectors built as described above. If nlinearly independent injection vectors are used the responses can beaggregated into new matrices V(s) and I(s) as follows:

$\begin{matrix}{{{V(s)}\overset{\Delta}{=}\begin{bmatrix}{v_{d\; 1}(s)} & {v_{q\; 1}(s)} \\\vdots & \vdots \\{v_{dn}(s)} & {v_{qn}(s)}\end{bmatrix}}{{I_{L}(s)}\overset{\Delta}{=}\begin{bmatrix}{i_{{Ld}\; 1}(s)} & {i_{{Lq}\; 1}(s)} \\\vdots & \vdots \\{i_{Ldn}(s)} & {i_{Lqn}(s)}\end{bmatrix}}} & (20)\end{matrix}$

Using these matrices as definitions, the impedance can be calculated as:

Z _(L)(S)=[I _(L)(S)^(T) I _(L)(S)]⁻¹ I _(L)(S)^(T) V(S)  (21)

This procedure can be repeated for the source impedance, replacingmatrix I_(L) with the corresponding matrix I_(s). When there are onlytwo samples used, the approaches are identical:

[I _(L)(S)^(T) I _(L)(S)]⁻¹ I _(L)(S)⁻¹ I _(L)(S)^(T-1) I _(L)(S)^(T) =I_(L)(S)⁻¹

Z _(L)(S)=I _(L)(S)^(T) V(S)  (22)

In order to measure a system's transfer function, it is necessary toobserve an input and an output of the system which are related by thetransfer function that defines the system. There may be signals presentat the point of measurement that are not created or considered by thesystem under test. It is important not to take these signals intoaccount as part of the calculation of a transfer function. Examples ofsuch signals are noise, sensor characteristics, and line frequencyharmonics.

Commercial equipment is available which, if included in the solution,allows for these details to be neglected by the algorithm as mentionedearlier. An example of such equipment is an Agilent 4395A Gain/Phaseanalyzer.

A set of software and hardware have been built to implement theimpedance measuring methodology presented. A schematic representation offunctional elements and their interrelationship of this implementationof the invention is illustrated in FIG. 5 to implement the algorithm andmethodology of the invention. As alluded to above, the synchronizationof the D-q frame with the input power for conversion of the ABCvariables of FIG. 1 to the D-q domain variables shown in FIG. 2 iscontrolled by phase locked loop 51 while the selection and control ofFrequency sweeps is performed by computer 52. The perturbation signal ofthe logic analyzer 59 is passed through anti-aliasing filters prior toA/D conversion. Once converted into a digital signal, the perturbationsignal follows two channels scaled by the d-q frame gains IpdGain andIqGain to form the resultant perturbation vector in the d-q frame. Thisvector is then converted to ABC variables at 57, and converted with D/Aconverter 55 into analog signals used as a reference for amplifiers 56which are preferably coupled to the system under test throughtransformers. Current and voltage sensors 58 a and 58 b measure theresponses to the perturbation sweeps and return measurements to thesystem through anti-aliasing filters and A/D converters for compensationof the perturbations in a manner not critical to the practice of theinvention as well as gain adjustment. The resultant conversion of theABC variables to the d-q domain for the two equivalent channels areperformed by elements 53 a and 53 b and multiplexed through multiplexer54 and converted into analog signals by the D/A converter 55 to providethe response signal to the impedance analyzer 59. This operation closesthe signal flow loop.

In the preferred embodiment of the invention hardware comprises a set ofthree amplifiers used to perform the injections and capable of producingperturbations of substantial power, sensors, transformers used toisolate the system, analog signal processing, and digital logic. Thehardware suitable for practice of the invention may be any of a widevariety of forms, the details of which are not at all critical to thepractice of the invention other than to provide current and voltageinjection and measurement capabilities suitable to the power transfercapacity of the system to be tested. However, the preferred hardwarearchitecture as described above is considered to be important.

The preferred architecture discussed above follows algorithm and systeminterface requirements. It consists of injection amplifiers used toinject the perturbation into the system, coupling magnetics, current andvoltage sensors for each phase, and a control system which is used toinject the disturbance. A prototype of the system is shown in FIG. 6.The coupling of the system in accordance with the invention to anarbitrary system under test is schematically illustrated in FIG. 7 andsequentially performs the measurement sweeps as illustrated in FIG. 7Ato gather the data from which the transfer functions of the phases ofthe ABC model of FIG. 1 can be determined.

To verify the operation of the algorithm, a passive load was connectedon the load side, and a source with resistors was connected on thesource side. The schematic of the system is shown in FIG. 7. The systemwas run up to the maximum current for the given source. The componentand operating point parameters are given in Table 1.

TABLE 1 System Under Test Parameters Description Value Unit RMS LineCurrent 5 A Line Frequency 500 Hz Load inductance 216 mH Loadcapacitance 90 mF Load resistance 3 Ω Source resistance 0.3 WTo verify the impedance, the stationary frame impedance was measured viaa precision impedance analyzer (HP 4194) and transformed into the D-qdomain. The transformed impedance and the corresponding measurement areshown in FIG. 8 and FIG. 9. It is to be noted that the measured dataextends past the computed impedances in the D-q domain because thecomputation method requires information about the stationary frameimpedance one line frequency below lowest computed D-q frequency.

In view of the foregoing, it is seen that the invention provides athree-phase impedance analyzer for the purposes of measuring three phaseimpedances in the d-q frame. A review of existing methods indicates thatwhile components of the total solution exist, one cohesive solution thataddresses all critical issues simultaneously does not. The inventionuses a conventional impedance analyzer and provides a suitable digitalcontrol system infrastructure built around it to enable the measurementof three-phase impedances in AC electrical systems The inventionachieves this function by injecting sequential perturbations on the d-qframe after proper alignment of the frame gas been achieved andmeasuring the response of the system in terms of the AC interfacevoltages and source and load currents. These inherently multi-phasevariables are converted into the d-q frame and transmitted back to theimpedance analyzer in a series of sequential measurements. Themeasurements are then used to solve the equations shown in FIG. 10,finally yielding the impedance values. The algorithm and methodologydescribed above allows for the measurement of three phase impedances inthe D-q frame via injection on multiple axes which are synchronized tothe three phase network via a variable bandwidth PLL. The acquisitionand analysis segments of the process have been described with supportingexperimental results. A hardware architecture has been introduced whichsupports the algorithm and power level requirements while sustainingcompatibility with the system under test.

While the invention has been described in terms of a single preferredembodiment, those skilled in the art will recognize that the inventioncan be practiced with modification within the spirit and scope of theappended claims.

Having thus described my invention, what I claim as new and desire tosecure by Letters Patent is as follows:
 1. Apparatus for measuringimpedances of respective phases of a multi-phase electrical power systemat any system interface between stages of said multi-phase electricalpower system, said apparatus comprising a phase locked loop for aligninga frame of reference with an input power vector and a plurality ofangles, a sweep generator for applying perturbations to respective phaseof said interface over a range of frequency, in sequence, to saidmulti-phase electrical power system, voltage and current sensors formeasuring amplitude and phase of the voltage and current responses torespective perturbations, and a computer to compute impedances ofrespective phases of said interface from phase and amplitude of voltageand current responses to said perturbations over said range of frequencyof respective said perturbations.
 2. The apparatus as recited in claim1, wherein said perturbations are produced by amplifiers.
 3. Theapparatus as recited in claim 1 wherein said perturbations have amagnitude in the range of 1% to 2% of the rated power of saidmulti-phase electrical power system.
 4. The apparatus as recited inclaim 1, wherein said perturbations are coupled to said multi-phaseelectrical power system through transformers.
 5. The apparatus asrecited in claim 1, wherein said perturbations are applied at differentangles in the d-q frame.
 6. The apparatus as recited in claim 5, whereinsaid perturbations are aligned with the d-axis and q-axis in the d-qframe.
 7. The apparatus as recited in claim 1, wherein said perturbationis converted to a three-phase variable prior to being applied to saidmulti-phase electrical power system
 8. The apparatus as recited in claim1 wherein said multi-phase electrical power system is a three-phaseelectrical power system.
 9. The apparatus as recited in claim 1 whereinsaid measurement of current and voltage includes measurement of bothamplitude and phase at each frequency in said range of frequencies ofsaid perturbations provided by said sweep generator as transferfunctions from which said impedances are calculated.
 10. The apparatusas recited in claim 1, wherein each perturbation is a vector and atleast two sets of vectors are applied.
 11. The apparatus as recited inclaim 1, wherein said voltages and currents are measure with respect toa repeatable reference which remains the same for all sweeps.
 12. Theapparatus as recited in claim 1, further including a digital controllerand an impedance analyzer.